Cross-ownership is a phenomenon where companies own parts of other companies they do business with. An example:

Two companies are now involved in the diamond operation, the mining group Anglo-American, and De Beers. The two are umbilically linked, making them impregnable to corporate raids. Some reshuffling has gone on of late but not much has changed: De Beers owns 42 per cent of Anglo- American and a similar share of its capital is owned by the mining group.

Assume that asset prices are the net present value of all future dividends, i.e. they contain no bubbles. (Incredible assumption, I know.)
Suppose Anglo-American changes a minor operating procedure and as a result will save a small amount of money each year, hence the net present value of its profits increase by \$1. This should increase the company value (and hence stock price) of Anglo-American, but since De Beers owns parts of the company it will share the increased cashflow and its company value (and hence stock price) should increase as well, increasing the company value of Anglo-American again, etc.

If I wanted to calculate the exact change in company value would I be correct in assuming I would have to create some sort of ownership matrix:
$$
M =
\begin{bmatrix}
58\% & 42\% \\
42\% & 58\%
\end{bmatrix}
$$
and take the Leontief inverse of this?
$$
\begin{bmatrix}
\Delta p_{AA} \\
\Delta p_{DB}
\end{bmatrix}
= (E-M)^{-1} \begin{bmatrix}
1 \\
0
\end{bmatrix}
$$

What if the Leontief inverse of the ownership matrix does not exist? E.g. both companies own half of the other. Is there any way to proceed in this case? (Note: I am not sure if it was possible to price companies with such an ownership structure to begin with.)

1 Answer
1

The way I would think of this is as follows. Let us write the value
of the first company as $V_{A}$and the second as $V_{B}.$ Given
your definition, let me know if you agree with the following argument:
$$
V_{A}=s_{AA}\sum_{t=0}^{\infty}\pi_{tA}+s_{AB}\sum_{t=0}^{\infty}\pi_{tB}
$$
Where $s_{AA}$ denotes the time invariant share of company A that
comapny A owns and $s_{BA}$ denotes the share of the company B that
A owns. Moreover, $\pi_{t}=A_{t}-L_{t}$ or the equity of each company.
By a symmetric argument, we have that :
$$
V_{B}=(1-s_{AA})\sum_{t=0}^{\infty}\pi_{tA}+(1-s_{AB})\sum_{t=0}^{\infty}\pi_{tB}
$$

Now, consider company $A.$ Taking differentials:
$$
dV_{A}=s_{AA}\sum_{t=0}^{\infty}d\pi_{tA}+s_{AB}\sum_{t=0}^{\infty}d\pi_{tB}
$$
Ceteris Paribus,you claim that $\sum_{t=0}^{\infty}d\pi_{tA}=1$
and the other term equals 0. So what you have then is:
$$
dV_{A}=s_{AA}\times1
$$
and
$$
dV_{B}=(1-s_{AA})\times1
$$
The answer is fairly intuitive- the change in value of each firm is
the change in profits that one of the firms has experienced times
the share of the profitable firm owned by either.
If each firm owned half, then the change in the value of each firm for an increase in $1 would be half, since each firm would have half the claim on the firm whose profit has increased by a half.